Installing toolboxes and setting up the path.

You need to unzip these toolboxes in your working directory, so that you have toolbox_signal, toolbox_general and toolbox_graph in your directory.

For Scilab user: you must replace the Matlab comment '%' by its Scilab counterpart '//'.

Recommandation: You should create a text file named for instance numericaltour.sce (in Scilab) or numericaltour.m (in Matlab) to write all the Scilab/Matlab command you want to execute. Then, simply run exec('numericaltour.sce'); (in Scilab) or numericaltour; (in Matlab) to run the commands.

Bending Invariants

Bending invariants replace the position of the vertices in a shape \(\Ss\) (2-D or 3-D) by new positions that are insensitive
to isometric deformation of the shape. This defines a bending invariant signature that can be used for surface matching.

Bending invariant were introduced in [EladKim03]. A related method was developped for brain flattening in [SchwShWolf89]. This method is related to the Isomap algorithm for manifold learning [TenSolvLang03].

We assume that \(Ss\) has some Riemannian metric, for instance coming from the embedding of a surface in 3-D Euclidian space,
or by restriction of the Euclian 2-D space to a 2-D sub-domain (planar shape). One thus can compute the geodesic distance
\(d(x,x')\) between points \(x,x' \in \Ss\).

The bending invariant \(\tilde \Ss\) of \(\Ss\) is defined as the set of points \(Y = (y_i)_j \subset \RR^d\) that are optimized
so that the Euclidean distance between points in \(Y\) matches as closely the geodesic distance between points in \(X\), i.e.
\[ \forall i, j, \quad \norm{y_i-y_j} \approx d(x_i,x_j) \]

Multi-dimensional scaling (MDS) is a class of method that aims at computing such a set of points \(Y \in \RR^{d \times N}\)
in \(\RR^d\) such that \[ \forall i, j, \quad \norm{y_i-y_j} \approx \de_{i,j} \] where \(\de \in \RR^{N \times N}\) is a
input data matrix. For a detailed overview of MDS, we refer to the book [BorgGroe97]

In this tour, we apply two specific MDS algorithms (strain and stress minimization) with input \(\de_{i,j} = d(x_i,x_j)\).